Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 541 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 + 518\cdot 541 + 520\cdot 541^{2} + 441\cdot 541^{3} + 281\cdot 541^{4} +O\left(541^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 87 + 63\cdot 541 + 166\cdot 541^{2} + 541^{3} + 35\cdot 541^{4} +O\left(541^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 246 + 91\cdot 541 + 97\cdot 541^{2} + 177\cdot 541^{3} + 148\cdot 541^{4} +O\left(541^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 284 + 379\cdot 541 + 21\cdot 541^{2} + 301\cdot 541^{3} + 189\cdot 541^{4} +O\left(541^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 393 + 29\cdot 541 + 276\cdot 541^{2} + 160\cdot 541^{3} + 427\cdot 541^{4} +O\left(541^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
